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<H3><A NAME="SECTION03242500000000000000">
Other Factorizations</A>
</H3>

<P>
The <B><I>QL</I></B> and <B><I>RQ</I></B>&nbsp;<B>factorizations</B>
<A NAME="2788"></A><A NAME="2789"></A> are given by
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
A = Q \left( \begin{array}{c} 0 \\L \end{array} \right) ,
\quad \mbox{if $m \geq n$,}
\end{displaymath}
 -->


<IMG
 WIDTH="203" HEIGHT="54" BORDER="0"
 SRC="img122.gif"
 ALT="\begin{displaymath}
A = Q \left( \begin{array}{c} 0 \\ L \end{array} \right) ,
\quad \mbox{if $m \geq n$,}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
A = \left( \begin{array}{cc} 0 & R \end{array} \right) Q,
\quad \mbox{if $m \leq n$.}
\end{displaymath}
 -->


<IMG
 WIDTH="220" HEIGHT="37" BORDER="0"
 SRC="img123.gif"
 ALT="\begin{displaymath}
A = \left( \begin{array}{cc} 0 &amp; R \end{array} \right) Q,
\quad \mbox{if $m \leq n$.}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
These factorizations are computed by xGEQLF and xGERQF, respectively; they are
<A NAME="2798"></A><A NAME="2799"></A><A NAME="2800"></A><A NAME="2801"></A>
<A NAME="2802"></A><A NAME="2803"></A><A NAME="2804"></A><A NAME="2805"></A>
less commonly used than either the <B><I>QR</I></B> or <B><I>LQ</I></B> factorizations
described above, but have applications in, for example, the
computation of generalized <B><I>QR</I></B> factorizations&nbsp;[<A
 HREF="node151.html#lawn31">2</A>].
<A NAME="2807"></A><A NAME="2808"></A>

<P>
All the factorization routines discussed here (except xTZRQF and xTZRZF) allow
arbitrary <B><I>m</I></B> and <B><I>n</I></B>, so that in some cases the matrices <B><I>R</I></B> or <B><I>L</I></B> are
trapezoidal rather than triangular.
A routine that performs pivoting is provided only for the <B><I>QR</I></B> factorization.

<P>
<BR>
<DIV ALIGN="CENTER">

<A NAME="tabcompof"></A>
<DIV ALIGN="CENTER">
<A NAME="2810"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 2.9:</STRONG>
Computational routines for orthogonal factorizations</CAPTION>
<TR><TD ALIGN="LEFT">Type of factorization</TD>
<TD ALIGN="LEFT">Operation</TD>
<TD ALIGN="CENTER" COLSPAN=2>Single precision</TD>
<TD ALIGN="CENTER" COLSPAN=2>Double precision</TD>
</TR>
<TR><TD ALIGN="LEFT">and matrix</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
<TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
</TR>
<TR><TD ALIGN="LEFT"><B><I>QR</I></B>, general</TD>
<TD ALIGN="LEFT">factorize with pivoting</TD>
<TD ALIGN="LEFT">SGEQP3<A NAME="2822"></A></TD>
<TD ALIGN="LEFT">CGEQP3<A NAME="2823"></A></TD>
<TD ALIGN="LEFT">DGEQP3<A NAME="2824"></A></TD>
<TD ALIGN="LEFT">ZGEQP3<A NAME="2825"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">factorize, no pivoting</TD>
<TD ALIGN="LEFT">SGEQRF<A NAME="2826"></A></TD>
<TD ALIGN="LEFT">CGEQRF<A NAME="2827"></A></TD>
<TD ALIGN="LEFT">DGEQRF<A NAME="2828"></A></TD>
<TD ALIGN="LEFT">ZGEQRF<A NAME="2829"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">generate <B><I>Q</I></B></TD>
<TD ALIGN="LEFT">SORGQR<A NAME="2830"></A></TD>
<TD ALIGN="LEFT">CUNGQR<A NAME="2831"></A></TD>
<TD ALIGN="LEFT">DORGQR<A NAME="2832"></A></TD>
<TD ALIGN="LEFT">ZUNGQR<A NAME="2833"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">multiply matrix by <B><I>Q</I></B></TD>
<TD ALIGN="LEFT">SORMQR<A NAME="2834"></A></TD>
<TD ALIGN="LEFT">CUNMQR<A NAME="2835"></A></TD>
<TD ALIGN="LEFT">DORMQR<A NAME="2836"></A></TD>
<TD ALIGN="LEFT">ZUNMQR<A NAME="2837"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"><B><I>LQ</I></B>, general</TD>
<TD ALIGN="LEFT">factorize, no pivoting</TD>
<TD ALIGN="LEFT">SGELQF<A NAME="2838"></A></TD>
<TD ALIGN="LEFT">CGELQF<A NAME="2839"></A></TD>
<TD ALIGN="LEFT">DGELQF<A NAME="2840"></A></TD>
<TD ALIGN="LEFT">ZGELQF<A NAME="2841"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">generate <B><I>Q</I></B></TD>
<TD ALIGN="LEFT">SORGLQ<A NAME="2842"></A></TD>
<TD ALIGN="LEFT">CUNGLQ<A NAME="2843"></A></TD>
<TD ALIGN="LEFT">DORGLQ<A NAME="2844"></A></TD>
<TD ALIGN="LEFT">ZUNGLQ<A NAME="2845"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">multiply matrix by <B><I>Q</I></B></TD>
<TD ALIGN="LEFT">SORMLQ<A NAME="2846"></A></TD>
<TD ALIGN="LEFT">CUNMLQ<A NAME="2847"></A></TD>
<TD ALIGN="LEFT">DORMLQ<A NAME="2848"></A></TD>
<TD ALIGN="LEFT">ZUNMLQ<A NAME="2849"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"><B><I>QL</I></B>, general</TD>
<TD ALIGN="LEFT">factorize, no pivoting</TD>
<TD ALIGN="LEFT">SGEQLF<A NAME="2850"></A></TD>
<TD ALIGN="LEFT">CGEQLF<A NAME="2851"></A></TD>
<TD ALIGN="LEFT">DGEQLF<A NAME="2852"></A></TD>
<TD ALIGN="LEFT">ZGEQLF<A NAME="2853"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">generate <B><I>Q</I></B></TD>
<TD ALIGN="LEFT">SORGQL<A NAME="2854"></A></TD>
<TD ALIGN="LEFT">CUNGQL<A NAME="2855"></A></TD>
<TD ALIGN="LEFT">DORGQL<A NAME="2856"></A></TD>
<TD ALIGN="LEFT">ZUNGQL<A NAME="2857"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">multiply matrix by <B><I>Q</I></B></TD>
<TD ALIGN="LEFT">SORMQL<A NAME="2858"></A></TD>
<TD ALIGN="LEFT">CUNMQL<A NAME="2859"></A></TD>
<TD ALIGN="LEFT">DORMQL<A NAME="2860"></A></TD>
<TD ALIGN="LEFT">ZUNMQL<A NAME="2861"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"><B><I>RQ</I></B>, general</TD>
<TD ALIGN="LEFT">factorize, no pivoting</TD>
<TD ALIGN="LEFT">SGERQF<A NAME="2862"></A></TD>
<TD ALIGN="LEFT">CGERQF<A NAME="2863"></A></TD>
<TD ALIGN="LEFT">DGERQF<A NAME="2864"></A></TD>
<TD ALIGN="LEFT">ZGERQF<A NAME="2865"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">generate <B><I>Q</I></B></TD>
<TD ALIGN="LEFT">SORGRQ<A NAME="2866"></A></TD>
<TD ALIGN="LEFT">CUNGRQ<A NAME="2867"></A></TD>
<TD ALIGN="LEFT">DORGRQ<A NAME="2868"></A></TD>
<TD ALIGN="LEFT">ZUNGRQ<A NAME="2869"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">multiply matrix by <B><I>Q</I></B></TD>
<TD ALIGN="LEFT">SORMRQ<A NAME="2870"></A></TD>
<TD ALIGN="LEFT">CUNMRQ<A NAME="2871"></A></TD>
<TD ALIGN="LEFT">DORMRQ<A NAME="2872"></A></TD>
<TD ALIGN="LEFT">ZUNMRQ<A NAME="2873"></A></TD>
</TR>
<TR><TD ALIGN="LEFT"><B><I>RZ</I></B>, trapezoidal</TD>
<TD ALIGN="LEFT">factorize, no pivoting</TD>
<TD ALIGN="LEFT">STZRZF<A NAME="2874"></A></TD>
<TD ALIGN="LEFT">CTZRZF<A NAME="2875"></A></TD>
<TD ALIGN="LEFT">DTZRZF<A NAME="2876"></A></TD>
<TD ALIGN="LEFT">ZTZRZF<A NAME="2877"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">(blocked algorithm)</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">multiply matrix by <B><I>Q</I></B></TD>
<TD ALIGN="LEFT">SORMRZ<A NAME="2878"></A></TD>
<TD ALIGN="LEFT">CUNMRZ<A NAME="2879"></A></TD>
<TD ALIGN="LEFT">DORMRZ<A NAME="2880"></A></TD>
<TD ALIGN="LEFT">ZUNMRZ<A NAME="2881"></A></TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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